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  <div class="section" id="numpy-cross">
<h1>numpy.cross<a class="headerlink" href="#numpy-cross" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.cross">
<code class="sig-prename descclassname">numpy.</code><code class="sig-name descname">cross</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">b</em>, <em class="sig-param">axisa=-1</em>, <em class="sig-param">axisb=-1</em>, <em class="sig-param">axisc=-1</em>, <em class="sig-param">axis=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/core/numeric.py#L1421-L1616"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.cross" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the cross product of two (arrays of) vectors.</p>
<p>The cross product of <em class="xref py py-obj">a</em> and <em class="xref py py-obj">b</em> in <img class="math" src="../../_images/math/7a2125a786b4c46d91c2dfeeea981521ddbb54ae.svg" alt="R^3"/> is a vector perpendicular
to both <em class="xref py py-obj">a</em> and <em class="xref py py-obj">b</em>.  If <em class="xref py py-obj">a</em> and <em class="xref py py-obj">b</em> are arrays of vectors, the vectors
are defined by the last axis of <em class="xref py py-obj">a</em> and <em class="xref py py-obj">b</em> by default, and these axes
can have dimensions 2 or 3.  Where the dimension of either <em class="xref py py-obj">a</em> or <em class="xref py py-obj">b</em> is
2, the third component of the input vector is assumed to be zero and the
cross product calculated accordingly.  In cases where both input vectors
have dimension 2, the z-component of the cross product is returned.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>a</strong><span class="classifier">array_like</span></dt><dd><p>Components of the first vector(s).</p>
</dd>
<dt><strong>b</strong><span class="classifier">array_like</span></dt><dd><p>Components of the second vector(s).</p>
</dd>
<dt><strong>axisa</strong><span class="classifier">int, optional</span></dt><dd><p>Axis of <em class="xref py py-obj">a</em> that defines the vector(s).  By default, the last axis.</p>
</dd>
<dt><strong>axisb</strong><span class="classifier">int, optional</span></dt><dd><p>Axis of <em class="xref py py-obj">b</em> that defines the vector(s).  By default, the last axis.</p>
</dd>
<dt><strong>axisc</strong><span class="classifier">int, optional</span></dt><dd><p>Axis of <em class="xref py py-obj">c</em> containing the cross product vector(s).  Ignored if
both input vectors have dimension 2, as the return is scalar.
By default, the last axis.</p>
</dd>
<dt><strong>axis</strong><span class="classifier">int, optional</span></dt><dd><p>If defined, the axis of <em class="xref py py-obj">a</em>, <em class="xref py py-obj">b</em> and <em class="xref py py-obj">c</em> that defines the vector(s)
and cross product(s).  Overrides <em class="xref py py-obj">axisa</em>, <em class="xref py py-obj">axisb</em> and <em class="xref py py-obj">axisc</em>.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>c</strong><span class="classifier">ndarray</span></dt><dd><p>Vector cross product(s).</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>ValueError</strong></dt><dd><p>When the dimension of the vector(s) in <em class="xref py py-obj">a</em> and/or <em class="xref py py-obj">b</em> does not
equal 2 or 3.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.inner.html#numpy.inner" title="numpy.inner"><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner</span></code></a></dt><dd><p>Inner product</p>
</dd>
<dt><a class="reference internal" href="numpy.outer.html#numpy.outer" title="numpy.outer"><code class="xref py py-obj docutils literal notranslate"><span class="pre">outer</span></code></a></dt><dd><p>Outer product.</p>
</dd>
<dt><a class="reference internal" href="numpy.ix_.html#numpy.ix_" title="numpy.ix_"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ix_</span></code></a></dt><dd><p>Construct index arrays.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.9.0.</span></p>
</div>
<p>Supports full broadcasting of the inputs.</p>
<p class="rubric">Examples</p>
<p>Vector cross-product.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array([-3,  6, -3])</span>
</pre></div>
</div>
<p>One vector with dimension 2.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array([12, -6, -3])</span>
</pre></div>
</div>
<p>Equivalently:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array([12, -6, -3])</span>
</pre></div>
</div>
<p>Both vectors with dimension 2.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array(-3)</span>
</pre></div>
</div>
<p>Multiple vector cross-products. Note that the direction of the cross
product vector is defined by the <em class="xref py py-obj">right-hand rule</em>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array([[-3,  6, -3],</span>
<span class="go">       [ 3, -6,  3]])</span>
</pre></div>
</div>
<p>The orientation of <em class="xref py py-obj">c</em> can be changed using the <em class="xref py py-obj">axisc</em> keyword.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">axisc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([[-3,  3],</span>
<span class="go">       [ 6, -6],</span>
<span class="go">       [-3,  3]])</span>
</pre></div>
</div>
<p>Change the vector definition of <em class="xref py py-obj">x</em> and <em class="xref py py-obj">y</em> using <em class="xref py py-obj">axisa</em> and <em class="xref py py-obj">axisb</em>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">array([[ -6,  12,  -6],</span>
<span class="go">       [  0,   0,   0],</span>
<span class="go">       [  6, -12,   6]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">axisa</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">axisb</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([[-24,  48, -24],</span>
<span class="go">       [-30,  60, -30],</span>
<span class="go">       [-36,  72, -36]])</span>
</pre></div>
</div>
</dd></dl>

</div>


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